The most memorable moments in anime often come when a character hits their low point. The worst thing that they can possibly imagine just happened to them - their loved one died, they lost their job, their depression got the better of them - whatever it is, they just can't function properly anymore.
Some anime characters who hit rock bottom actually bounce back pretty quickly - Simon is devastated by Kamina's death in Gurren Lagann, but he comes back from it stronger than ever. In other cases, it's a downward slide into oblivion - Taizou Hasegawa is never going to stop being a MADAO, and Sayaka Miki's downfall in Puella Magi Madoka Magica results in her death. Whatever the case, seeing characters at their lowest can be a great way to reveal a new aspect of their personalities - or a great way to make yourself sad.
After Tomoya's wife Nagisa dies during childbirth, he completely loses his motivation to do anything. He starts drinking heavily, stops going to work, and worst of all, he completely neglects to play a role in his newborn daughter Ushio's life. Nagisa's parents - who are also grieving the loss of their daughter - step in to take care of Ushio while Tomoya gets his act together, but it takes him nearly five years. When he finally does try to reinsert himself into his daughter's life, it's too late - by this point, Ushio has developed the same condition that plagued her mother and is about to die herself. It's this particular conflict that makes CLANNAD one of the saddest anime in history.
After his little sister Ayumi dies while trying to defend herself from a classmate with a box cutter, Yuu Otosaka loses all motivation to care about anything. He abandons his efforts to help other young people with supernatural powers and starts spending every day playing video games in an Internet cafe, eating junk food, and getting into fights. He actually finds some crack cocaine - something that is rarely seen in anime - and is about to start smoking it when Nao Tomori, his friend and future love interest - appears to stop him. After that, he starts getting his act together and goes back to school and back to the student council.
To say that Simon took Kamina's death hard would be an understatement. He's so devastated by it that he holes up in his room, refusing to talk to anyone or continue team Dai Gurren's mission. Instead, he spends all his time crying and carving statues of Kamina. It's only when Nia appears to give him a brutally honest pep talk that he actually pulls himself out of his misery and declares himself to be a separate person from his idol, who is ready to stand on his own two feet. Unlike some of these tragic downfalls, Simon's doesn't last forever, and he emerges from it stronger than before.
Shinji Ikari has some pretty good reasons for acting out - his father is forcing him to pilot a giant death machine against his will, and talking to him like he's a particularly disgusting insect the whole time. He's deeply depressed, has few meaningful connections with anyone in his life, and he's terrified of what the future will bring.
One of the actions he ultimately takes to express his inner turmoil is disgusting and disrespectful - but it certainly shows how far down the hole he is. When his fellow Eva pilot Asuka is comatose in the hospital, he masturbates onto her unconscious body. It's hard to get much lower than that.